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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000294
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000294: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
St000294: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => 00 => 3
[1,2,4,3] => [1,2,3] => 00 => 3
[1,3,2,4] => [1,3,2] => 01 => 4
[1,3,4,2] => [1,3,2] => 01 => 4
[1,4,2,3] => [1,2,3] => 00 => 3
[1,4,3,2] => [1,3,2] => 01 => 4
[2,1,3,4] => [2,1,3] => 10 => 4
[2,1,4,3] => [2,1,3] => 10 => 4
[2,3,1,4] => [2,3,1] => 10 => 4
[2,3,4,1] => [2,3,1] => 10 => 4
[2,4,1,3] => [2,1,3] => 10 => 4
[2,4,3,1] => [2,3,1] => 10 => 4
[3,1,2,4] => [3,1,2] => 10 => 4
[3,1,4,2] => [3,1,2] => 10 => 4
[3,2,1,4] => [3,2,1] => 11 => 3
[3,2,4,1] => [3,2,1] => 11 => 3
[3,4,1,2] => [3,1,2] => 10 => 4
[3,4,2,1] => [3,2,1] => 11 => 3
[4,1,2,3] => [1,2,3] => 00 => 3
[4,1,3,2] => [1,3,2] => 01 => 4
[4,2,1,3] => [2,1,3] => 10 => 4
[4,2,3,1] => [2,3,1] => 10 => 4
[4,3,1,2] => [3,1,2] => 10 => 4
[4,3,2,1] => [3,2,1] => 11 => 3
[1,2,3,4,5] => [1,2,3,4] => 000 => 4
[1,2,3,5,4] => [1,2,3,4] => 000 => 4
[1,2,4,3,5] => [1,2,4,3] => 001 => 6
[1,2,4,5,3] => [1,2,4,3] => 001 => 6
[1,2,5,3,4] => [1,2,3,4] => 000 => 4
[1,2,5,4,3] => [1,2,4,3] => 001 => 6
[1,4,3,2,5] => [1,4,3,2] => 011 => 6
[1,4,3,5,2] => [1,4,3,2] => 011 => 6
[1,4,5,3,2] => [1,4,3,2] => 011 => 6
[1,5,2,3,4] => [1,2,3,4] => 000 => 4
[1,5,2,4,3] => [1,2,4,3] => 001 => 6
[1,5,4,3,2] => [1,4,3,2] => 011 => 6
[2,1,3,4,5] => [2,1,3,4] => 100 => 6
[2,1,3,5,4] => [2,1,3,4] => 100 => 6
[2,1,5,3,4] => [2,1,3,4] => 100 => 6
[2,3,4,1,5] => [2,3,4,1] => 100 => 6
[2,3,4,5,1] => [2,3,4,1] => 100 => 6
[2,3,5,4,1] => [2,3,4,1] => 100 => 6
[2,5,1,3,4] => [2,1,3,4] => 100 => 6
[2,5,3,4,1] => [2,3,4,1] => 100 => 6
[3,2,1,4,5] => [3,2,1,4] => 110 => 6
[3,2,1,5,4] => [3,2,1,4] => 110 => 6
[3,2,5,1,4] => [3,2,1,4] => 110 => 6
[3,4,2,1,5] => [3,4,2,1] => 110 => 6
[3,4,2,5,1] => [3,4,2,1] => 110 => 6
[3,4,5,2,1] => [3,4,2,1] => 110 => 6
Description
The number of distinct factors of a binary word.
This is also known as the subword complexity of a binary word, see [1].
Matching statistic: St000518
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000518: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000518: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => 00 => 3
[1,2,4,3] => [1,2,3] => 00 => 3
[1,3,2,4] => [1,3,2] => 01 => 4
[1,3,4,2] => [1,3,2] => 01 => 4
[1,4,2,3] => [1,2,3] => 00 => 3
[1,4,3,2] => [1,3,2] => 01 => 4
[2,1,3,4] => [2,1,3] => 10 => 4
[2,1,4,3] => [2,1,3] => 10 => 4
[2,3,1,4] => [2,3,1] => 01 => 4
[2,3,4,1] => [2,3,1] => 01 => 4
[2,4,1,3] => [2,1,3] => 10 => 4
[2,4,3,1] => [2,3,1] => 01 => 4
[3,1,2,4] => [3,1,2] => 10 => 4
[3,1,4,2] => [3,1,2] => 10 => 4
[3,2,1,4] => [3,2,1] => 11 => 3
[3,2,4,1] => [3,2,1] => 11 => 3
[3,4,1,2] => [3,1,2] => 10 => 4
[3,4,2,1] => [3,2,1] => 11 => 3
[4,1,2,3] => [1,2,3] => 00 => 3
[4,1,3,2] => [1,3,2] => 01 => 4
[4,2,1,3] => [2,1,3] => 10 => 4
[4,2,3,1] => [2,3,1] => 01 => 4
[4,3,1,2] => [3,1,2] => 10 => 4
[4,3,2,1] => [3,2,1] => 11 => 3
[1,2,3,4,5] => [1,2,3,4] => 000 => 4
[1,2,3,5,4] => [1,2,3,4] => 000 => 4
[1,2,4,3,5] => [1,2,4,3] => 001 => 6
[1,2,4,5,3] => [1,2,4,3] => 001 => 6
[1,2,5,3,4] => [1,2,3,4] => 000 => 4
[1,2,5,4,3] => [1,2,4,3] => 001 => 6
[1,4,3,2,5] => [1,4,3,2] => 011 => 6
[1,4,3,5,2] => [1,4,3,2] => 011 => 6
[1,4,5,3,2] => [1,4,3,2] => 011 => 6
[1,5,2,3,4] => [1,2,3,4] => 000 => 4
[1,5,2,4,3] => [1,2,4,3] => 001 => 6
[1,5,4,3,2] => [1,4,3,2] => 011 => 6
[2,1,3,4,5] => [2,1,3,4] => 100 => 6
[2,1,3,5,4] => [2,1,3,4] => 100 => 6
[2,1,5,3,4] => [2,1,3,4] => 100 => 6
[2,3,4,1,5] => [2,3,4,1] => 001 => 6
[2,3,4,5,1] => [2,3,4,1] => 001 => 6
[2,3,5,4,1] => [2,3,4,1] => 001 => 6
[2,5,1,3,4] => [2,1,3,4] => 100 => 6
[2,5,3,4,1] => [2,3,4,1] => 001 => 6
[3,2,1,4,5] => [3,2,1,4] => 110 => 6
[3,2,1,5,4] => [3,2,1,4] => 110 => 6
[3,2,5,1,4] => [3,2,1,4] => 110 => 6
[3,4,2,1,5] => [3,4,2,1] => 011 => 6
[3,4,2,5,1] => [3,4,2,1] => 011 => 6
[3,4,5,2,1] => [3,4,2,1] => 011 => 6
Description
The number of distinct subsequences in a binary word.
In contrast to the subword complexity [[St000294]] this is the cardinality of the set of all subsequences of not necessarily consecutive letters.
Matching statistic: St000926
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000926: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000926: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([],3)
=> 3
[1,2,4,3] => [1,2,3] => ([],3)
=> 3
[1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 4
[1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 4
[1,4,2,3] => [1,2,3] => ([],3)
=> 3
[1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 4
[2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 4
[2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 4
[2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> 4
[2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[3,1,4,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [1,2,3] => ([],3)
=> 3
[4,1,3,2] => [1,3,2] => ([(1,2)],3)
=> 4
[4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 4
[4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 4
[4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 4
[1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 4
[1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 6
[1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 6
[1,2,5,3,4] => [1,2,3,4] => ([],4)
=> 4
[1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 6
[1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 6
[1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 6
[1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 6
[1,5,2,3,4] => [1,2,3,4] => ([],4)
=> 4
[1,5,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 6
[1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 6
[2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 6
[2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 6
[2,1,5,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 6
[2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 6
[2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 6
[2,3,5,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 6
[2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 6
[2,5,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 6
[3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 6
[3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 6
[3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 6
[3,4,2,1,5] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
Description
The clique-coclique number of a graph.
This is the product of the size of a maximal clique [[St000097]] and the size of a maximal independent set [[St000093]].
Matching statistic: St001880
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,1,4,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,4,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[3,2,4,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[3,4,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,4,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3,5] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,2,4,5,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,4,3,2,5] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,4,3,5,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,4,5,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,5,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,5,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,1,3,4,5] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,1,3,5,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,1,5,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,3,4,1,5] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,3,4,5,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,3,5,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,5,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,5,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,2,1,4,5] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,2,1,5,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,2,5,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,4,2,1,5] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000108
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[1,3,2,4] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[1,3,4,2] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[1,4,3,2] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[2,1,3,4] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[2,1,4,3] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[2,3,1,4] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[2,3,4,1] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[2,4,1,3] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[2,4,3,1] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[3,1,2,4] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[3,1,4,2] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[3,2,1,4] => [3,2,1] => [3]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,1] => [3]
=> 4 = 3 + 1
[3,4,1,2] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[3,4,2,1] => [3,2,1] => [3]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[4,2,1,3] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[4,2,3,1] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[4,3,1,2] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[4,3,2,1] => [3,2,1] => [3]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,5,4,3,2] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,3,4,1,5] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[2,3,4,5,1] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[2,3,5,4,1] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,5,3,4,1] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[3,2,1,4,5] => [3,2,1,4] => [3,1]
=> 7 = 6 + 1
[3,2,1,5,4] => [3,2,1,4] => [3,1]
=> 7 = 6 + 1
[3,2,5,1,4] => [3,2,1,4] => [3,1]
=> 7 = 6 + 1
[3,4,2,1,5] => [3,4,2,1] => [3,1]
=> 7 = 6 + 1
[3,4,2,5,1] => [3,4,2,1] => [3,1]
=> 7 = 6 + 1
[3,4,5,2,1] => [3,4,2,1] => [3,1]
=> 7 = 6 + 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[1,3,2,4] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[1,3,4,2] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[1,4,3,2] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[2,1,3,4] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[2,1,4,3] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[2,3,1,4] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[2,3,4,1] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[2,4,1,3] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[2,4,3,1] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[3,1,2,4] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[3,1,4,2] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[3,2,1,4] => [3,2,1] => [3]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,1] => [3]
=> 4 = 3 + 1
[3,4,1,2] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[3,4,2,1] => [3,2,1] => [3]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2] => [2,1]
=> 5 = 4 + 1
[4,2,1,3] => [2,1,3] => [2,1]
=> 5 = 4 + 1
[4,2,3,1] => [2,3,1] => [2,1]
=> 5 = 4 + 1
[4,3,1,2] => [3,1,2] => [2,1]
=> 5 = 4 + 1
[4,3,2,1] => [3,2,1] => [3]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 4 + 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> 7 = 6 + 1
[1,5,4,3,2] => [1,4,3,2] => [3,1]
=> 7 = 6 + 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,3,4,1,5] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[2,3,4,5,1] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[2,3,5,4,1] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> 7 = 6 + 1
[2,5,3,4,1] => [2,3,4,1] => [2,1,1]
=> 7 = 6 + 1
[3,2,1,4,5] => [3,2,1,4] => [3,1]
=> 7 = 6 + 1
[3,2,1,5,4] => [3,2,1,4] => [3,1]
=> 7 = 6 + 1
[3,2,5,1,4] => [3,2,1,4] => [3,1]
=> 7 = 6 + 1
[3,4,2,1,5] => [3,4,2,1] => [3,1]
=> 7 = 6 + 1
[3,4,2,5,1] => [3,4,2,1] => [3,1]
=> 7 = 6 + 1
[3,4,5,2,1] => [3,4,2,1] => [3,1]
=> 7 = 6 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St001658
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => [3] => [[3],[]]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,3] => [3] => [[3],[]]
=> 4 = 3 + 1
[1,3,2,4] => [1,3,2] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[1,3,4,2] => [1,3,2] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[1,4,2,3] => [1,2,3] => [3] => [[3],[]]
=> 4 = 3 + 1
[1,4,3,2] => [1,3,2] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[2,1,3,4] => [2,1,3] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[2,1,4,3] => [2,1,3] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[2,3,1,4] => [2,3,1] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[2,3,4,1] => [2,3,1] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[2,4,1,3] => [2,1,3] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[2,4,3,1] => [2,3,1] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[3,1,2,4] => [3,1,2] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[3,1,4,2] => [3,1,2] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[3,2,1,4] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
=> 4 = 3 + 1
[3,4,1,2] => [3,1,2] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[3,4,2,1] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => [3] => [[3],[]]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[4,2,1,3] => [2,1,3] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[4,2,3,1] => [2,3,1] => [2,1] => [[2,2],[1]]
=> 5 = 4 + 1
[4,3,1,2] => [3,1,2] => [1,2] => [[2,1],[]]
=> 5 = 4 + 1
[4,3,2,1] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => [4] => [[4],[]]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4] => [4] => [[4],[]]
=> 5 = 4 + 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[1,2,4,5,3] => [1,2,4,3] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[1,2,5,3,4] => [1,2,3,4] => [4] => [[4],[]]
=> 5 = 4 + 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
[1,5,2,3,4] => [1,2,3,4] => [4] => [[4],[]]
=> 5 = 4 + 1
[1,5,2,4,3] => [1,2,4,3] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[1,5,4,3,2] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
[2,1,3,4,5] => [2,1,3,4] => [1,3] => [[3,1],[]]
=> 7 = 6 + 1
[2,1,3,5,4] => [2,1,3,4] => [1,3] => [[3,1],[]]
=> 7 = 6 + 1
[2,1,5,3,4] => [2,1,3,4] => [1,3] => [[3,1],[]]
=> 7 = 6 + 1
[2,3,4,1,5] => [2,3,4,1] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[2,3,4,5,1] => [2,3,4,1] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[2,3,5,4,1] => [2,3,4,1] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[2,5,1,3,4] => [2,1,3,4] => [1,3] => [[3,1],[]]
=> 7 = 6 + 1
[2,5,3,4,1] => [2,3,4,1] => [3,1] => [[3,3],[2]]
=> 7 = 6 + 1
[3,2,1,4,5] => [3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> 7 = 6 + 1
[3,2,1,5,4] => [3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> 7 = 6 + 1
[3,2,5,1,4] => [3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> 7 = 6 + 1
[3,4,2,1,5] => [3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
[3,4,5,2,1] => [3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 7 = 6 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000520
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000520: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000520: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => [3,2,1] => 4 = 3 + 1
[1,2,4,3] => [1,2,3] => [3,2,1] => 4 = 3 + 1
[1,3,2,4] => [1,3,2] => [3,1,2] => 5 = 4 + 1
[1,3,4,2] => [1,3,2] => [3,1,2] => 5 = 4 + 1
[1,4,2,3] => [1,2,3] => [3,2,1] => 4 = 3 + 1
[1,4,3,2] => [1,3,2] => [3,1,2] => 5 = 4 + 1
[2,1,3,4] => [2,1,3] => [2,3,1] => 5 = 4 + 1
[2,1,4,3] => [2,1,3] => [2,3,1] => 5 = 4 + 1
[2,3,1,4] => [2,3,1] => [2,1,3] => 5 = 4 + 1
[2,3,4,1] => [2,3,1] => [2,1,3] => 5 = 4 + 1
[2,4,1,3] => [2,1,3] => [2,3,1] => 5 = 4 + 1
[2,4,3,1] => [2,3,1] => [2,1,3] => 5 = 4 + 1
[3,1,2,4] => [3,1,2] => [1,3,2] => 5 = 4 + 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 5 = 4 + 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 4 = 3 + 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 4 = 3 + 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 5 = 4 + 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 4 = 3 + 1
[4,1,2,3] => [1,2,3] => [3,2,1] => 4 = 3 + 1
[4,1,3,2] => [1,3,2] => [3,1,2] => 5 = 4 + 1
[4,2,1,3] => [2,1,3] => [2,3,1] => 5 = 4 + 1
[4,2,3,1] => [2,3,1] => [2,1,3] => 5 = 4 + 1
[4,3,1,2] => [3,1,2] => [1,3,2] => 5 = 4 + 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => [4,3,2,1] => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4] => [4,3,2,1] => 5 = 4 + 1
[1,2,4,3,5] => [1,2,4,3] => [4,3,1,2] => 7 = 6 + 1
[1,2,4,5,3] => [1,2,4,3] => [4,3,1,2] => 7 = 6 + 1
[1,2,5,3,4] => [1,2,3,4] => [4,3,2,1] => 5 = 4 + 1
[1,2,5,4,3] => [1,2,4,3] => [4,3,1,2] => 7 = 6 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,1,2,3] => 7 = 6 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,1,2,3] => 7 = 6 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,1,2,3] => 7 = 6 + 1
[1,5,2,3,4] => [1,2,3,4] => [4,3,2,1] => 5 = 4 + 1
[1,5,2,4,3] => [1,2,4,3] => [4,3,1,2] => 7 = 6 + 1
[1,5,4,3,2] => [1,4,3,2] => [4,1,2,3] => 7 = 6 + 1
[2,1,3,4,5] => [2,1,3,4] => [3,4,2,1] => 7 = 6 + 1
[2,1,3,5,4] => [2,1,3,4] => [3,4,2,1] => 7 = 6 + 1
[2,1,5,3,4] => [2,1,3,4] => [3,4,2,1] => 7 = 6 + 1
[2,3,4,1,5] => [2,3,4,1] => [3,2,1,4] => 7 = 6 + 1
[2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 7 = 6 + 1
[2,3,5,4,1] => [2,3,4,1] => [3,2,1,4] => 7 = 6 + 1
[2,5,1,3,4] => [2,1,3,4] => [3,4,2,1] => 7 = 6 + 1
[2,5,3,4,1] => [2,3,4,1] => [3,2,1,4] => 7 = 6 + 1
[3,2,1,4,5] => [3,2,1,4] => [2,3,4,1] => 7 = 6 + 1
[3,2,1,5,4] => [3,2,1,4] => [2,3,4,1] => 7 = 6 + 1
[3,2,5,1,4] => [3,2,1,4] => [2,3,4,1] => 7 = 6 + 1
[3,4,2,1,5] => [3,4,2,1] => [2,1,3,4] => 7 = 6 + 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 7 = 6 + 1
[3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 7 = 6 + 1
[8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
[1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
[1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
[1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 + 1
Description
The number of patterns in a permutation.
In other words, this is the number of subsequences which are not order-isomorphic.
Matching statistic: St001664
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001664: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St001664: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 4 + 1
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 4 + 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 4 + 1
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 5 = 4 + 1
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 5 = 4 + 1
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> 5 = 4 + 1
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 5 = 4 + 1
[3,1,4,2] => [3,1,2] => ([(1,2)],3)
=> 5 = 4 + 1
[3,2,1,4] => [3,2,1] => ([],3)
=> 4 = 3 + 1
[3,2,4,1] => [3,2,1] => ([],3)
=> 4 = 3 + 1
[3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 5 = 4 + 1
[3,4,2,1] => [3,2,1] => ([],3)
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 5 = 4 + 1
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 5 = 4 + 1
[4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 5 = 4 + 1
[4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 5 = 4 + 1
[4,3,2,1] => [3,2,1] => ([],3)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 7 = 6 + 1
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 7 = 6 + 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 7 = 6 + 1
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7 = 6 + 1
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7 = 6 + 1
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7 = 6 + 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 7 = 6 + 1
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 7 = 6 + 1
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 7 = 6 + 1
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 7 = 6 + 1
[2,1,5,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 7 = 6 + 1
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 7 = 6 + 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 7 = 6 + 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 7 = 6 + 1
[2,5,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 7 = 6 + 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 7 = 6 + 1
[3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7 = 6 + 1
[3,2,1,5,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7 = 6 + 1
[3,2,5,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7 = 6 + 1
[3,4,2,1,5] => [3,4,2,1] => ([(2,3)],4)
=> 7 = 6 + 1
[3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 7 = 6 + 1
[3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 7 = 6 + 1
[8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
[1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
[1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
[1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 + 1
Description
The number of non-isomorphic subposets of a poset.
Matching statistic: St000419
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,3] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,3,2,4] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[1,3,4,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[1,4,2,3] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,4,3,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[2,1,3,4] => [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[2,1,4,3] => [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[2,3,1,4] => [2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[2,3,4,1] => [2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[2,4,1,3] => [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[2,4,3,1] => [2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[3,1,2,4] => [3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[3,1,4,2] => [3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[3,2,1,4] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,4,1,2] => [3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[3,4,2,1] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[4,1,2,3] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[4,2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[4,2,3,1] => [2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[4,3,1,2] => [3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 4
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,3,5,4] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,4,3,5] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[1,2,4,5,3] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[1,2,5,3,4] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,5,4,3] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[1,5,2,3,4] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,5,2,4,3] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[2,1,3,4,5] => [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,1,3,5,4] => [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,1,5,3,4] => [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,3,4,1,5] => [2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,3,4,5,1] => [2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,3,5,4,1] => [2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,5,1,3,4] => [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[2,5,3,4,1] => [2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[3,4,2,1,5] => [3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[3,4,2,5,1] => [3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[3,4,5,2,1] => [3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6
[8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000420The number of Dyck paths that are weakly above a Dyck path. St000087The number of induced subgraphs. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001645The pebbling number of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001623The number of doubly irreducible elements of a lattice. St001625The Möbius invariant of a lattice. St001754The number of tolerances of a finite lattice. St000058The order of a permutation.
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